Abstract
UniMath, short for “Univalent Mathematics”, refers to both a language (a.k.a. a formal system) for mathematics, as well as to a computer-checked library of mathematics formalized in that system. The UniMath library, under active development, aims to coherently integrate machine-checked proofs of mathematical results from many different branches of mathematics.
The UniMath language is a dependent type theory, augmented by the univalence axiom. One goal is to keep the language as small as possible, to ease verification of the theory. In particular, general inductive types are not part of the language.
In this work, we partially remedy this lack by constructing some inductive (families of) sets. This involves a formalization of a standard category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the inductive sets thus obtained.
The present work constitutes one part of a construction of a framework for specifying, via a signature, programming languages with binders as nested datatypes. To this end, we are going to combine our work with previous work by Ahrens and Matthes (itself based on work by Matthes and Uustalu) on an axiomatisation of substitution for languages with variable binding. The languages specified via the framework will automatically be equipped with the structure of a monad, where the monadic operations and axioms correspond to a well-behaved substitution operation.
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References
Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carol. 15(4), 589–602 (1974)
Ahrens, B., Capriotti, P., Spadotti, R.: Non-wellfounded trees in homotopy type theory. In: Altenkirch, T. (ed.) 13th International Conference on Typed Lambda Calculi and Applications, TLCA, Warsaw, Poland, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 1–3 July 2015. LIPIcs, vol. 38, pp. 17–30 (2015)
Ahrens, B., Kapulkin, K., Shulman, M.: Univalent categories and the Rezk completion. Math. Struct. Comput. Sci. 25, 1010–1039 (2015)
Ahrens, B., Matthes, R.: Heterogeneous substitution systems revisited. CORR, abs/1601.04299 (2016)
Awodey, S., Gambino, N., Sojakova, K.: Inductive types in homotopy type theory. In: Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, Dubrovnik, Croatia, 25–28 June 2012, pp. 95–104. IEEE Computer Society (2012)
Bezem, M., Coquand, T., Huber, S.: A model of type theory in cubical sets. In: Matthes, R., Schubert, A. (eds.) 19th Conference on Types for Proofs and Programs, TYPES 2013, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. LIPIcs, vol. 26, pp. 107–128 (2013)
Bird, R.S., Paterson, R.: De Bruijn notation as a nested datatype. J. Funct. Prog. 9(1), 77–91 (1999)
Cohen, C., Coquand, T., Huber, S., Mörtberg, A.: Cubical Type Theory: a constructive interpretation of the univalence axiom (2015, Preprint)
Gonthier, G., Mahboubi, A.: An introduction to small scale reflection in Coq. J. Formaliz. Reason. 3(2), 95–152 (2010)
Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)
Matthes, R., Uustalu, T.: Substitution in non-wellfounded syntax with variable binding. Theor. Comput. Sci. 327(1–2), 155–174 (2004)
Voevodsky, V.: An experimental library of formalized mathematics based on the univalent foundations. Math. Struct. Comput. Sci. 25, 1278–1294 (2015). http://arxiv.org/pdf/1401.0053.pdf
Acknowledgments
We thank Dan Grayson, Ralph Matthes, Paige North and Vladimir Voevodsky for helpful discussions on the subject matter.
This material is based upon work supported by the National Science Foundation under agreement Nos. DMS-1128155 and CMU 1150129-338510. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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Ahrens, B., Mörtberg, A. (2016). Some Wellfounded Trees in UniMath. In: Greuel, GM., Koch, T., Paule, P., Sommese, A. (eds) Mathematical Software – ICMS 2016. ICMS 2016. Lecture Notes in Computer Science(), vol 9725. Springer, Cham. https://doi.org/10.1007/978-3-319-42432-3_2
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DOI: https://doi.org/10.1007/978-3-319-42432-3_2
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